on matsushima's formula for the betti numbers of a ... - Project Euclid

Hotta, R. and Wallach, N.R. Osaka J. Math. 12 (1975), 419-431

ON MATSUSHIMA'S FORMULA FOR THE BETTI NUMBERS OF A LOCALLY SYMMETRIC SPACE RYOSHI HOTTA AND NOLAN R. WALLACH (Received September 11, 1974) (Revised December 9, 1974) 1. Introduction. Let G be a connected semi-simple Lie group with finite center and with no connected normal, compact subgroups. Let KdG be a maximal compact subgroup and let Γ C G be a discrete subgroup acting freely on X=G/K and so that Γ\G is compact. Let M=T\X then M is a typical compact locally symmetric space of negative curvature. Let G act by the right regular action, πΓy on L2(Γ\G). In Matsushima [13] a formula for the Betti numbers of M is given in terms of the multiplicities of certain unitary representations of G in πr. In this paper we investigate the existence of the representations of G that come into the Matsushima formula. In particular we show that if X is irreducible and Hermitian symmetric and if rank X>p then there are no unitary representations of G that satisfy the conditions for the Matsushima formulas for the (0, p) Betti number of T\X, b0>p(T\X). Thus we find that if pp(T\X)=0. In particular if rank X>1 we recover Matsushima's theorem [12] that the first Betti number of Γ\^Γ is zero. Actually this result (on the first Betti number) follows from the more general theorem of Kazhdan which says if G is a simple Lie group of split rank larger than 1 and if ΓcG is a discrete subgroup so that Γ\G has finite volume relative to some Haar measure on G then Γ/[Γ, Γ] is finite. In light of the above result of Kazhdan it is reasonable to study the unitary representations that appear in the formula of Matsushima for the first Betti number in the case G has split rank 1. We show that in this case such representations always exist. We show that there are at most two such unitary representations and if G is locally SO(n, 1) there is exactly one, let us call it π^ (see Lemma 2.1, Prop. 2.2 and Lemma 4.4). This gives us some interesting examples. E. B. Vinberg [16] has constructed a uniform discrete subgroup TdSOe(ny 1) for 3 t>c) is reducible and |)c=|>+φp- with (T, £+) and (T, £-) irreducible (in this case G is locally isomorphic with SU(n, 1) for some n). (2) (r, pc) is irreducible. In case (1) we take TO to be (T, p~) in case (2) we take TO to be (T, pc). Let Ω be the Casimir operator of g. That is, if B is the Killing form of g, if X19 •••, Xn is a basis of g and if Yί9 •••, Yn satisfy B(Xi9 Yj)=8fj' then Ω— Lemma 2.1. There exists an irreducible unitary representation (π^ H^) of G so that (π^fζi τQ)=l and τr1(Ω)=0. (Here if σly σz are representations of K then (σι: σ2) is the dimension of the space of K-intertwίnίng operators from σί to σ2). Proof. In Kostant [10] (see also Johnson-Wallach [7]) it is shown that if π is a (non-unitary) prinicipal series representation of G and if σ is an irreducible unitary representation of K then (π\ κ: σ)= 1 or 0 depending on whether or not σ has an M-fixed vector. In Johnson-Wallach [7] it is shown if (τr°, V) is the representation of G on the space V of ^-finite, C°° functions on G/MAN (MAN =P a minimal parabolic subgroup of G, M=K Π P, K/M^G/P) defined by (π\X)f) (x) =

at

f(^(-tX}x) I ,_ for /e

then every constituent of the (finite) composition series extends to a unitary representation of G. Now TO has an M-fixed vector thus (τr°|κ : τ 0 )=l. Hence exactly one of the irreducible constituents of π° contains TO. We denote this constituent by π,. Clearly τr°(Ω)=0 (1 e V). Thus ^(Ω)=0. Proposition 2.2.

Suppose that G is in case (1) above or that G is locally

MATSUSHIMA'S FORMULA

421

ίsomorphic with SOe(ny 1), n>3 (the identity component of the group of the quadratic form ΣϊîΛ?— #S+1). V(π> H) is an irreducible unitary representation of G so that (π\κ: τ 0 )φO and τr(Ω)=0 then π is unitarily equivalent with π^ Proof. The assertion in case (1) will be proved in the next section. We therefore assume that G is locally isomorphic with SOe(n, 1)=G0. If (TT, H) contains TO then since the center of G, Z, is contained in K and τQ(Z)= {/} we see that π(Z)=I. We may therefore assume that G=G0. Let P=MAN be a minimal parabolic subgroup of G such that M=P Π K and the Lie algebra of A, α, is contained in p. Let Mbe the set of equivalence classes of irreducible unitary representations of M and let α*c be the space of complex valued linear forms on α. If tΐ is the Lie algebra of N then there is a unique element, Jf/eα, so that acLHΊn=/. Now a=RH. If ξ eΛfr and z/eα* c let (ττ f ι V , H**) be the corresponding (non-unitary) prinicipal series representation of G0. That is H^ is the space of all /: G-*H% ((ξ , H$) is a representative of ξ) such that / is measurable and (i) f(gman)=ξ(mΓe~^°**>f(g) for g^G, mv , (T O | M : f)φO. Now K acts on \> as 5O(w) on /2W. M is just SO(n— 1) imbedded in ^ as Γ SO(n-l) 0 ] L 0 1 J. Thus if ξ0 is the standard representation of M on CM 1 and 1 is the trivial representation of M, T O | M =ί 0 φl. Thus ^r must be a constituent of τr l f V for some v or τr£ 0 f V for some v. If τr 1>v (Ω)=0 then v(H) =Q or w— 1. Since τr 1 > 0 and TT^^.! have the same composition series we see that if π is a constituent of τr 1 > v for some v then 7r is equivalent to π±. If τr£ o>v (Ω)—0 then v(H)—z or n—#—1 for some .sreC by the usual formula for ;rw(Ω): ^ ίθiV (Ω) = const. where c(ξ0)^C is a constant depending only on f 0 . τr^ 0 > w _ 2 _ 1 and ^ô 2 have the same character hence the same composition series. Thus since (T O | M : f 0 )=l

422

R. HOTTA AND N.R. WALLACH

there is at most one other irreducible unitary representation of G satisfying the required conditions and it must appear in πtQtίg. We show that the constituent of τrg 0 t 2 containing TO is infinitesimally equivalent to πl and thereby prove the proposition. Let 7\ be a maximal torus of M. Then TÂ is a Cartan subgroup of G0. Let Gc=SO(n-\-\y C) and let CdGc be the Cartan subgroup corresponding to TÂ. Let £) be the Lie algebra of C in gc. Let Δ be the root system of (gc, Ij) and let Δ+ be a system of positive roots such that if α£ΞΔ and a(//)>0 then α>0. Then if rt+—SÂ'Sαπ rt + Πg—π. Let #ι, •••, α/ be the corresponding simple roots (/ is the largest integer tf/ we maY assume α2, •••, α/eΔ M Let vQ^Hξo=Cn~l be a non-zero highest weight vector and let w0 be a lowest vector. If f^H^ set *S(/) (#)— ((Z, W)=^ZiWi). If /eC-(G) define for *e$c, ^=^+^2, 4 (/te expίÔ+ί/fe expί^2)I ί=0. If ?0(ίK=λ(ίK for all αί fo »then (I) RxS(f) = Q for Jϊen + (II) S(f) (gtά) = \(t)e-^^S(f) (g) for ίe Γ,, aÂ. The Borel-Weil theorem (c.f. Wallach [17]) implies that S is a 1-1 map from the space of C°° elements of H*'*, Hjj,o'v, onto the space of all/e C°°(G) satisfying (I), (II). Let .AΓλ'v be the space of all C°° functions on G satisfying (I), (II). If ft=X* \x9gv(g)oS. λ v Hence the representations (T λfV JSΓ ' ) and (τr^ θ j V , Jϊβ!0>v) of g are infinitesimally equivaldent. Let now tfêί)^-{0} (g_ αι = {êgc| [A, ^J^-α^)^ for all Set for /eifc 0 , B(f)(g)=(RE_cύJ) (g). A direct computation gives c^f λ>v with z/=α 1 |α. But J5 is clearly non-zero. Hence v=z or w—^—1. Furthermore, it is shown in Johnson-Wallach [7] that £P'°/(constants) is irredu1>0 cible. Thus ( π ί 9 ί/j) is infinite simally equivalent with (τr1>0, ί/ /constants) and by the above is contained in π $ Q t Z . This completes the proof of the Proposition. NOTES. 1. If G satisfies (2) then one can use the same sort of arguments as those proving Proposition 2.2 to show that there are at most 2 irreducible unitary representations, π, of G satisfying (π\κ: τ 0 )Φθ and τr(Ω)=0. 2. We note that if π is irreducible (π\κ: 1)ΦO and τr(Ω)—0 then π is the trivial representation of G. Indeed, the subquotient theorem implies π is a constituent of τr 1>v for some ι>. But ττ ljV (Ω)=0 implies z/=0 or z;—2p. Both contain the trivial representation. Using a similar argument we prove


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Proposition 2.3. Let G be locally isomorphic with SU(n,l) and n>2. Then G satisfies (1). As a representation space of K, $+®$~=Cv0ξ&(τ1>ly Vltl) with (Ad(*)® Ad(ft)) (VQ)=VO, kl, Hltl) of G so that (2) If n=2 and if π is an irreducible unitary representation of G so that (π\κ\ τ M )φO and τr(Ω)=0 then π is unitarily equivalent with π1'1. Furthermore π1'1 is a non-integrableί squeare integrable representation of G. Proof. We assume (as in the proof of Proposition 2.2) that G—SU(n, 1). The statements about t>+®JΓ are standard. Again we look at the composition series of τr 1>0 and find that there is a unique constituent, π1'1 containing TM. Since every constituent of τr l f 0 is unitarizable (1) follows. To prove (2) take G=SU(2,l). It is proved in Johnson-Wallach [7] that 7Γ1'1 is non-integrable discrete series. As a representation of M, r1 >x — 1®^®^, ξ{ characters of M. MA. is a Cartan subgroup of G. Ordering the roots as in the proof of Proposition 2.1 the simple roots that are not roots of M consists of «!, α2 (say). One finds H1^Q/Ktr(RE_cύl)+KGτ(RE_cύ2) is the representation space of π1'1. The proof of uniqueness now follows as in the proof of Proposition 2.2. NOTE. (2) above is actually true for all SU(n, 1), n>2 using a similar argument to the proof of (2) above. Here one must use RE-ΛI^ E-ΛZ ôr an appropriate k> 1 to "pick up" the extra representation of M. These "extra intertwining operators" will be studied systematically in another paper. 3.

Certain representations with highest weights

Let G be a connected, simply connected, simple Lie group. Let g be the Lie algebra of G and let g— ϊφp be a Cartan decomposition of g (see Helgason [3] chapters 3 and 8 for the pertinent definitions). Let QC be the complexification of g. We assume that [ϊ, ϊ]Φϊ. Let £)# be a maximal abelian subalgebra of ϊ. If rj is the complexification of ξ* in QC then ΐ) is a Cartan subalgebra of gc.Let Δ be the root system of gc relative to fj. We may (and do) assume that ϊ=ΛιH'®[I, ϊ] with a(H)=±l or 0 for αeΔ. Let Δ^^ {a e Δ \ g* C ϊc} , Δ P = {a e Δ \ gΛ C pc} . Here lc and pc are the respective complexifications of f and $ in gc. Then Δ^— {a e Δ \ a(H)=Q} . Let f)R=if)^={h^\a(h)^R for all αeΔ}. Let H=H19 H2, .-, Ht be a basis of ήΛ. Order ΐ)Jl3Δ lexicographically relative to the ordered basis Hlt •••, Hj. Let Δ+ be the corresponding system of positive roots for Δ. Set Δ^=Δ+ +

+

n Δ X ) Δ £ = Δ n Δ P . Set t) =Σ-SΛίβ. f"=Σ.eΔίfl-. Thenp c =ί) ± ± + and zaH\^—I, ad//| p - = — /. Hence ad (ϊc) p cϊ> . Also set n =Σα!

424


DEFINITION 3.1. A representation (TT, V) of gc is said to have highest weight Ae5*=5J®C if there is VQ^ V, ^ 0 ΦO so that (1) τr(#K=Λ(#K forffefc (2) ar(n>0=0, (3) π(U(Qc))vQ=V. Here C/(gc) is the universal enveloping algebra of gc. DEFINITION 3.2. A representation (π, V) of gc is said to be Q-infinitesimally unitary if there exists a positive definite inner product < , > on V such that , w>+;, 7r(JQtt;>=0 for all Λ"eg, v, WSΞ V. Theorem 3.3. (Harish-Chandra [2]). If(π, V) is a Q-infinitesimally unitary representation of gc with highest weight Λ and if V is the Hubert space completion of (V,(y) then there exists an irreducible unitary representation σ of G on V so that the differential of σ restricted to V is π. Proof. The only difference between the above statement and Theorem 4 of Harish-Chandra [2] is the irreducibility statement. Let τ;0e V be as in Definition 3.1. It is easy to see that if v^ V and π(h)v=A(h)v for all A e ΐ ) then v=cv0 for some scalar c. Suppose that WdV is an invariant subspace. Then W^ is invariant. Clearly v0^ W or Ôe W^. Hence W— V or W^= V. Thus π is indeed irreducible if it is g-infinitesimally unitary. Lemma 3.4. Let β be the largest root of Δ. If Λeϊj* and =0 « , > is the dual in bilinear form on t)* to B on t}) and if Λ Φ O then no representation with highest weight Λ can be Q-infinitesimally unitary. Proof. Let gM— ϊφφ and let Eêg^ define a Weyl basis of gc/ί) relative to gw. That is if T is the conjugation of gc relative to Qu(τ(X-}-iY)=;X—iY, X, Yeg M ) then rEΛ=-E.Λ and [EΛ, E.^HΛ with β(//ΓJ, H)=a(H) for ffeή. If (TT, V) is g-infinitesimally unitary relative to < , > then ζπ(X)v, wy=—= -Δ(HP)= -=0 (since w( Now — /β is the lowest weight of (ad, gc). Hence ad([/(n+))£'_β=gc. (t/(n+) the universal enveloping algebra of n+). On the other hand 0=π(n+U(n+))π (E_β)v0 = π(ad(n+U(n+)) E_β)v τλ denote the irreducible finite dimensional representation of ΐc with highest weight λ (c.f. Wallach [18], chapter 4). (1) If s^ Wc then sρ—ρ is ^-dominant integral if and only ifs&W1. (2) σft=]2ίeτrι> TSP-P Proof. (1) If s^Wc and ^0 for all #eΔ£ then >0 for all αeΔ£. Thus αeίΔ+ for all α^Δ£. Let α be a simple root in Δ£ (this is equivalent to 2===-=)= Σ det(ί) «•«*', «p*c» Π (l-e-"tt>)= Σ det(ί)es ΛeΔ +

êϊr^,

«eΔs:+

*eWs

(here ρκ=~- 5] tf) (see for example Wallach [18], Lemma 4.9.5.). 2 ΛeΔκ +

Hence,

Now σk(cxpt H)=ektτ. Thus the representations Σθσ2Aί and of pc are disjoint. Also if σk— ^ nk(\)rλy then the Weyl character formula (c.f. Wallach [18], 4.9.6.) says ( Σ det (*)*"*) (X + -X-) = Σ(-l)*Σ»*(λ) Σ

det (5) Thus we see A

λ

seir^

From this, the above observations and (1), we see that wA(λ)=0 or 1 and all of the highest weights appearing with wΛ(λ)=t=0 must be of the form sρ—p, $e

426


Wl. Now, if s^ Wc, sp— -p=Σ Ύ the sum taken over the elements of sΔ+ Π Δ". If SΪΞ Wl then jΔ+ Π Δ~cΔp= {—a I αeΔJ} . (a) If s^W1 and if αeΔJ, TesΔ+ΠΔ" then if α+γeΔ, α+7 δ+μk and if s^ Wl with l(s)2. Let jy be the Weyl reflection about the hyperplane a~ 0

Let γn •••, jr be elments of Δp so that (a) γôf! (i) ifιΦy,7ί±7 y ΦΔ (t) r is maximal subject to (a), (b). Then r— split rank (G) (See Harish-Chandra [3]). (d) Let ^"=20/^7,.. Then if αeΔ^, α|r is of one of the following forms: -=-(7,.— 7^), i>j, — -^-7^, 0. If αeΔp then α|ίr is of one of the 1 1 following forms -y (7y+7y), i2, a£k+1\^-=—^-7t or —(K=( | λ+

Now if a^Δp then (π(E^v, w^=— , π(E_^wy, vy w^ domain {π(EΛ), π(E_^} . Using this we see that if we take the inner product of the left hand side of (3) with vμ we have

(4) -2ΣU 4 ,+IW£«Mlλ+Pl*-l/H-p|*).

But every highest weight of Fλ®Λ^~ is of the form λ— where QcΔp is a subset and =Σ-6βα. But if 0CΔ+, |λ+p|2- |λ-+p| 2>0. (See Kostant [9]). Hence 7r(/?Λ)ϋμ=0 for αeΔp. This clearly implies π has highest weight μ. Q.E.D. NOTE: Case (1) of Proposition 2.2 follows from Lemma 3.7 using λ=0 since £~~ is irreducible and thus there is only one possible μ with Wj(0: μ)Φθ. Corollary 3.8. Suppose that split rank G>&. // k>0 and if Σ^Cμ)^- then there does not exist a unitary irreducible representation (π, H) ofGso that (1) τr(Ω)=0 (2) Proof. If such a (π, H) existed then it would have to have highest weight μ for some μ with w*(μ)ΦO. But then μ=sp—ρ for some se W71, l(s)—k. But then ΦO by Lemma 3.4 and Lemma 3.6 yields a contradiction. NOTE. The above result is best possible in light of Lemma 2.1. Since split rank SU(n, 1)— 1 and π± in the notation of Lemma 2.1 satisfies (1), (2) above with A = l .

4.

The Betti numbers of T\X

Let G be a connected, simple Lie group with finite center and let K be a maximal compact subgroup of G. Let Γ C G be a discrete subgroup of G so that Γ has no elements of finite order and Γ\G is compact. Let X=G/K. Then T\X is a compact locally symmetric space. Let g be the Lie algebra of G and let g— f ®t> be the Cartan decomposition of g corresponding to K. Let dg be a Haar measure on G and let d'g denote the corresponding G invariant measure on Γ\G. Let πr be the regular representation of G on L2(Γ\G). Then it is well known that as a representation of G, πΓ = ^ω^NΓ(ω)ω (here ό is the set of equivalence classes of irreducible unitary representations of G) and Nτ(ω) < oo . Let (ΛÂdl/r), Λ^c)— Σ w />,λ τ λ with τ λ the irreducible unitary representation of K with highest weight λ. If [ϊ, ϊ] Φ ϊ then ^c^t^θp" as a representation of ^ and let (Λ*Ad| *


429

τ λ the irreducible representation of K with highest weight X. Theorem 4.1. (Matsushima [13], Matsushima-Murakami [14]). (1) bp(Γ\X)=^^QNΓ(π)^mp^0(π\κ:rλ). (2) // [f, f]φϊ then T\X is a Kάhler manifold and bp>q(Γ\X)^^^0Nτ(π)^λ>nptq.^0(π\κ: rλ). Here (2)

above combined with Corollary 3.8 immediately implies

Theorem 4.2.

// rank X=k then b0tg(T\X)=Q for 00(Γ/X). But T\X is Kahler. We also note (using the notation of §2)

bί>0(T\X)=b0>l(Γ\X)

since

Lemma 4.4. If G is locally isomorphίc with SU(n, 1) then bl(T\X)= 2Nτ(πl}. If G is locally isomorphic with SOe(n, 1) andn>3 then b1(Γ\X)=NΓ(π1). Otherwise if G is of split rank 1 then We conclude this paper by looking at the example G=SU(2,l). Let us denote, πly by 7Γ0>1. Let Λ be the highest weight of Λ2p~~ . Then the irreducible representation of G with highest weight Λ (in the sense of §3) is a holomorphic discrete series representation. It is not integrable however so Langlands' formula [11] does not apply. (It can also be shown that the formula of HottaParthasarathy [6] does not apply either). Denote this representation by τr°'2. Let 7Γ1'1 be as in Proposition 2.3. Then b0>1(T\X)=Nτ(π°'1)9 bΰt2(T\X)=NΓ(π°'2)9 b, l(Γ\X)=NΓ(πltl)+l. If we normalize dg so that vol (Γ\G)=f

JT/G

d'g=X(Γ\X)

the Euler number of T\X then the Max Noether formula combined with the Hirzebruch proportionality principle implies (α) 1 -έ0,1(Γ\Jί)+ό0>2(Γ\/^)=-l vol (Γ\G).

also χ(r\*)=ΣM-iWr\^). Poincarέ duality implies (b) 2-261(Γ\^)+ό2(Γ\^)=vol (Γ\G). Hence we have vol (Γ\G)

(a') l-N^Ϊ+NrW)

3-4NΓ(π ' )+2NΓ(π' ' )+NΓ(π1 1)=vol (Γ\G). Now since vol(Γ\G)>0 we see that if b0ι2(T\X)=Q then ό0,1(Γ\^Γ)=0 hence vol(Γ\G)=3. Hence b2(Γ\X)=l. Thus (as is well known in this case) if Γ is such that NΓ(π0f2)=0 then the real cohomology ring of Γ\X is generated by the (V)

0 1

> z

430


Kahler class and is isomorphic with the real cohomology ring of CP2. We also note that (β7), (V] combined imply NΓ(π^}=Nτ( We can see no reason in the harmonic analysis of L2(Γ\G) why this should be true. Finally we note that in (#'), τr°>2 is discrete series but π°ίl is a so called "trash representation". That is, it is non-tempered but its character has support in the elliptic regular elements. Paul Sally and the authors have named these representations trash because they seem to be the reason why the expected formulae for the multiplicities of discrete series are not right. That is, if the trash has been disposed of the formula is correct. Note that the trivial representation is a trash representation. Similar coupling of non-integrable discrete series and trash representations have been found by Paul Sally and the second author of this paper for the twofold cover of SL(2, R). HIROSHIMA UNIVERSITY RUTGERS UNIVERSITY

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manifolds, J. Differential Geometry 1 (1967), 99-109. , S. Murakami: On certain cohomology groups attached to Hermitian symmetric spaces I, II, Osaka J. Math. 2 (1965), 365-416, 5 (1968), 223-241. 2 [15] R. Parthasarathy: A note on the vanishing of certain "L cohomologies", J. Math. Soc. Japan 23 (1971), 676-691. [16] E.B. Vinberg: Some examples of crystallographic groups on Lobάcevskii spaces, Mat. Sb. 78 (1969), 633-639; Math. USSR-Sb. 7 (1969), 617-622. [17] N.R. Wallach: Induced representations of Lie algebras and a theorem of BorelWeil, Trans. Amer. Math. Sec. 136 (1969), 181-187. [18] : Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, Inc., New York, 1973. [19] G. Warner: Harmonic Analysis on Semi-simple Lie Groups II, SpringerVerlag, New York, 1972. [14]